How does $s=r\theta$ take into account negative angles or angles greater than $2\pi$? Radian is defined as the angle subtended by the arc at the center of the circle.
The formula of radian is $\theta=\dfrac sr$ where $s$ is the arc length and $r$ is the circle's radius.
Since length of arc cannot be more than the circumference, how is a value of radian that is more than $2\pi$ adjusted in the formula?
How about a negative value of radian?
Is a scalar $k$ multiplied in the formula to compensate for negative value and for value more than to $2\pi\;?$ Is this formula correct? $$\theta=k\frac sr$$
 A: You can continue using the $$s=r\theta$$ formula even when the angular displacement $\theta$ rad is negative or exceeds $2\pi$ rad: while the radius $r$ is positive, the signed distance $s$ travelled along the circumference is negative if and only if the travel direction is clockwise.
To elaborate: think of a thread spooling around its reel, where the thread is winding around the reel multiple times; if spooling is considered to be in the positive direction, then unspooling is considered to be in the negative direction.

Addendum (migrating up my comment replies)

*

*$s$ is not displacement.
Measuring turns


*$s$ and $\theta$ are scalars, not vectors in 3D space (though they can be framed as one-dimensional vectors).
They do not have free direction in 3D space; their direction is conceptual.
Scalars and Euclidean vectors
A: Imagine walking around the circle a distance of $l$. If $l$ is positive, you walk clockwise. If $l$ is negative, you walk counterclockwise. If $l$ is more than $2\pi$, you walk around the circle multiple times. Then, if your initial location is $A$, the circle has center $O$ and radius $r$, and your final location is $B$, then you could say that angle $AOB$ is $l/r$ radians.
Adding $2\pi$ doesn’t affect things, so you generally add or subtract that until you get a number in the range $0$ to $2\pi$. If you’re given an angle to begin with, and calculate the radian using the circumference, you’ll always get a number in that range.
